How Do Trigonometric Identities Simplify Complex Equations?

[ritwik06]

Homework Statement

If $$\frac{(cos x)^{4}}{(cos y)^{2}}+\frac{(sin x)^{4}}{(sin y)^{2}}=1$$ prove that


$$\frac{(cos y)^{4}}{(cos x)^{2}}+\frac{(sin y)^{4}}{(sin x)^{2}}=1$$

The Attempt at a Solution

$$(cos x)^{4} (sin y)^{2}+(sin x)^{4} (cos y)^{2}=(sin y)^{2}-(sin y)^{4}$$


On simplification:
$$\frac{(sin y)^{4}}{(sin x)^{2}}=(sin y)^{2}+ (cos x)^{2} (sin y)^2 - (sin x)^{2}(cos y)^{2}$$


Similarly for cos
$$\frac{(cos y)^{4}}{(cos x)^{2}}=(cos y)^{2}+ (cos y)^{2} (sin x)^2 - (cos x)^{2}(sin y)^{2}$$


Adding the above gives the result.

But is their any simpler way?

 

[morphism]

Am I missing something, because both your identities are identical?!

 

[ritwik06]

Am I missing something, because both your identities are identical?!

There is nothing missing. Please check it out once again, they are indeed different.